A study on anabelian geometry of higher local fields

Abstract

Anabelian geometry has been developed over a much wider class of fields than Grothendieck, who is the originator of anabelian geometry, conjectured. So, it is natural to ask the following question: What kinds of fields are suitable for the base fields of anabelian geometry? In this paper, we consider this problem for higher local fields. First, to consider “anabelianness” of higher local fields themselves, we give mono-anabelian reconstruction algorithms of various invariants of higher local fields from their absolute Galois groups. As a result, the isomorphism classes of certain types of higher local fields are completely determined by their absolute Galois groups. Next, we prove that mixed-characteristic higher local fields are Kummer-faithful. This result affirms the above question for these higher local fields to a certain extent.